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A novel method for the automatic evaluation of retinal vessel tortuosity Enrico Grisan, Marco Foracchia and Alfredo Ruggeri Enrico Grisan, Marco Foracchia and Alfredo Ruggeri Abstract Tortuosity is among the first alterations in retinal vessel network to appear in many retinopathies. Automatic evaluation of retinal vessel tortuosity is thus a valuable tool for early detection of vascular suffering. Quite a few techniques for its measurement and classification have been proposed, but they do not always match the clinical concept of tortuosity. This justifies the need for a new definition, able to express in mathematical terms the tortuosity as perceived by ophthalmologists. We propose here a new algorithm for the evaluation of tortuosity in vessels extracted from digital fundus images. It is based on the partitioning of each vessel in segments of constant-sign curvature and on the combination between the number of such segments and their curvature values. This algorithm has been compared with the other tortuosity measures on a set of 20 vessels from 10 different images. These vessels had been preliminarily ordered by an expert ophthalmologist in order of increasing perceived tortuosity. The proposed algorithm proved to be the best one with regards to arterial tortuosity and among the best for vein tortuosity evaluation. Introduction Results and discussion Methods Available Tortuosity Measures Proposed Tortuosity Measure Available Data Acknowledgements This work was partially supported by a research grant form Nidek Technologies, Italy. The authors would like to thank Prof. S. Piermarocchi and colleagues at the Department of Ophthalmology, University of Padova, for providing images and clinical advice. M. Foracchia is now with M 2 Scientific Computing, Italy. Bibliography [1] W. Hart et al., Int. J. Med. Inf., 53, 239-252, 1999 [2] C. Heneghan et al., Med. Imag. An., 6, 407-429, 2002 [3] M. E. Martinez-Perez et al., IEEE Trans. Biom. Eng., 20, 1193-1200, 2002 [4] K. V. Chandrinos et al., Proc. MEDICON98, 1998 [5] M. Foracchia et al., CAFIA 2001, 15, 2001 Many diseases have the retina as target organ, and some are only recognizable by changes in the vascular network of the retina. One of the first changes is the increase in vessel tortuosity:  Tortuosity measure has to match the clinical perception of ophthalmologists  Understand the factors that influence the classification of a vascular structure as tortuous or non-tortuous.  In retinal images, straight vessels but also long vessels presenting a smooth semi- circular shape are considered as non-tortuous. Previously proposed methods (see [1] for a review) failed in differentiating the tortuosity of structures that visually appeared to be very different in tortuosity, as it will be shown. Tortuosity Properties A set of 20 vessels from 10 different retinal images was used. Images were acquired with a fundus camera with a 50 o field of view and then digitized; vessel segments were automatically extracted by a previously developed tracking algorithm [5], and sorted by increasing tortuosity by an expert ophthalmologist (arteries and veins separately ) A vessel is a continous curve in a two dimensional space, and it can be described by a sampled version of it. Sampling of a vessel may lead to a very sparse vessel description:  Poor information on vessel direction and its derivatives  Noisy sample direction information Cubic smoothing spline fit:  describes the vessel between sampling points, where no data are present.  give a C 1 (at least) representation of the vessel (physiologically sound description) Affine Transformation Composition Modulation  Geographical position in the retina does not affect tortuosity perception: translation invariant  Vessel orientation does not affect tortuosity perception: rotation invariant  Scale does not seem to affect tortuosity perception, but this is controversial. The evaluation of a single vessel tortuosity might be considered invariant to scaling Adjacent continuous curves s 1 and s 2, the combination of the two is: In [1], an intuitive principle was proposed for the tortuosity of the composition:  Fig. 2 top panel, shows that this statement can not be accepted in conjunction with the principle of invariance with respect to rotation and scale: connecting three non-tortuous curves yields an undoubtedly tortuous curve.  New composition property, such that a vessel s, combination of various segments s i, will not have tortuosity measure less than any of its composing parts:  For two vessels having twists (changes in curvature sign) with the same amplitude (maximum distance of the curve from the underlying chord), the difference in tortuosity varies with the number of twists: frequency modulation.  For two vessels with the same number of twists (with the same frequency), the difference in tortuosity depends on the difference in amplitude of the twists: amplitude modulation Arc Length over length ratio Measure involving curvature Mean direction angle change Ratio between its length and the length of the underlying chord The greater the value of the ratio, the more distant the vessel is from a straight line, i.e., tortuous [1][2][3]  Being the surface of the retina close to a semi-sphere, the non tortuous paradigm should be the circle arc  Fig. 2 (top and middle panel) shows that two vessels with the same arc/chord ratio may have very different perceived tortuosity Various positive functions of the curvature [1]: curvature should be a measure of the variability of vessel direction.  The curve in Fig. 2 middle panel :integral of the absolute value of curvature is π  The curve in Fig. 2 bottom panel :integral of the absolute value of curvature is π/2  The bottom panel curve has greater perceived tortuosity  Composition of straight lines and arcs dramatically changes tortuosity perception  Changes in curvature sign are not taken into account  Integral value depends upon the integration domain Average of difference in vessel directions among samples within a distance window [4]  Sensitivity to noise  Difference in direction fails to account for changes in convexity Figure 1 Vessel centerline and borders sample (left panel), and vessel samples interpolation through cubic spline (right panel) Figure 2 Tortuosity measures counterexamples (see text) Integrating information  Number of times a vessel changes the curvature sign (twist)  Amplitude of each twist 1]Being C s (l) the curvature of segment s, described by the curvilinear coordinate l belonging to the domain D, each vessel s is partitioned in a set of n subsegments s i, i=1,..., n such that:  This represents an hysteretic thresholding of the curvature (Fig. 3)  Each subsegment has a quasi-constant curvature sign (the quasi given by the hysteresis) 2]For each subsegment s i the arc length ratio R i is computed 3]Total vessel length L, its tortuosity becomes: Figure 3 Vessel subsegment evaluation, based on the hyteretic thresholding of the curvature The proposed tortuosity measure:  Is invariant to rotation and translation  Composition is fulfilled through the summation  Amplitude modulation is accounted for with R i  Frequency modulation is accounted for through vessel subdivision (implicitly), and through the multiplication by n-1 (explicitly)  Division by L makes the tortuosity measures indipendent from scaling.  The proposed measures achieved a correlation of 0.870 for the artey vessel set (best among all the measures proposed in [1],[2],[4])  The proposed measures achieved a correlation of 0.870 for the vein vessel set (third best among all the measures proposed in [1],[2],[4])